Symmetry in the Green's function for birth-death chains

TL;DR

This paper investigates a symmetry property of the Green's function in birth-death chains, providing two proofs—one using Brownian motion local time and the other electric network reciprocity—and extends the second proof to chains on trees.

Contribution

It introduces a new symmetry relation for the Green's function in birth-death chains and offers two distinct proofs, including an extension to chains on trees.

Findings

Symmetry relation in the Green's function for birth-death chains.

Two proofs: one via Brownian motion local time, another via electric network reciprocity.

Extension of the second proof to birth-death chains on trees.

Abstract

A symmetric relation in the probabilistic Green's function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green's functions in other contexts. The first uses as primary tool the local time of Brownian motion, while the second uses the reciprocity principle from electric network theory. We also show that the the second proof extends easily to cover birth-death chains (a.k.a. state-dependent random walks) on trees.